Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

TL;DR

This paper introduces a stable space-time tensor Galerkin method for low-rank approximation of linear parabolic equations, utilizing a minimal residual formulation and iterative greedy algorithms, with promising numerical results.

Contribution

It presents a novel stable Galerkin approach for low-rank space-time approximation of parabolic equations using tensor methods and minimal residual formulation.

Findings

Method is stable uniformly in discretization parameters.

Numerical results demonstrate effectiveness on complex operators.

Comparison shows advantages over fully discrete Petrov-Galerkin methods.

Abstract

We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a stable Galerkin method, uniformly in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert--Bochner spaces. The discrete solution is sought in a trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performances of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov--Galerkin setting to evaluate the dual residual norm.